| import OAI.Analysis.FreeEntropy.EntropyDeficit |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Set Filter Finset |
| namespace FiniteEntropySeparation.Model |
| variable (p : ℕ) [NeZero p] |
| theorem reciprocal_variance_intervalIntegrable (i : Fin (p+1)) (a b : ℝ) |
| (ha : 0 < a) (hab : a ≤ b) : |
| IntervalIntegrable (fun u => (variance p i + u)⁻¹) volume a b := by |
| apply ContinuousOn.intervalIntegrable |
| apply ContinuousOn.inv₀ (continuous_const.add continuous_id).continuousOn |
| intro u hu |
| rw [uIcc_of_le hab] at hu |
| change variance p i + u ≠ 0 |
| exact ne_of_gt (by linarith [variance_nonneg p i, hu.1]) |
| theorem covarianceFisher_intervalIntegrable (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| IntervalIntegrable (covarianceFisher p) volume a b := by |
| change IntervalIntegrable (fun u => ∑ i : Fin (p+1), (variance p i + u)⁻¹) volume a b |
| simpa only [Finset.sum_fn] using (IntervalIntegrable.sum univ (fun i _ => reciprocal_variance_intervalIntegrable p i a b ha hab)) |
| theorem covarianceFisher_integral (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| (∫ u in a..b, covarianceFisher p u) = |
| ∑ i : Fin (p+1), (Real.log (b + variance p i) - Real.log (a + variance p i)) := by |
| unfold covarianceFisher |
| rw [intervalIntegral.integral_finsetSum (fun i _ => reciprocal_variance_intervalIntegrable p i a b ha hab)] |
| apply sum_congr rfl |
| intro i _ |
| rw [intervalIntegral.integral_comp_add_left (fun u : ℝ => u⁻¹), integral_inv_of_pos |
| (by linarith [variance_nonneg p i]) (by linarith [variance_nonneg p i]), |
| Real.log_div (by linarith [variance_nonneg p i] : variance p i+b ≠ 0) |
| (by linarith [variance_nonneg p i] : variance p i+a ≠ 0)] |
| simp only [add_comm] |
| theorem gaussianEntropy_flow (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| gaussianEntropy p b - gaussianEntropy p a = |
| (1/2 : ℝ) * ∫ u in a..b, covarianceFisher p u := by |
| rw [covarianceFisher_integral p a b ha hab, sum_sub_distrib] |
| unfold gaussianEntropy |
| ring |
| theorem fisherDeficit_intervalIntegrable (hp : 2 ≤ p) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| IntervalIntegrable (fisherDeficit p) volume a b := by |
| rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab] |
| exact (fisherDeficit_tail_integrable p hp a ha).mono_set (fun u hu => hu.1) |
| theorem heatFisherReal_intervalIntegrable (hp : 2 ≤ p) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| IntervalIntegrable (heatFisherReal p) volume a b := by |
| have hh := (fisherDeficit_intervalIntegrable p hp a b ha hab).add |
| (covarianceFisher_intervalIntegrable p a b ha hab) |
| change IntervalIntegrable (fun x => (freeFisher (τ p) (X p x)).toReal) volume a b |
| simpa only [fisherDeficit, sub_add_cancel] using hh |
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| theorem chiStar_flow (hp : 2 ≤ p) (a b : ℝ) (ha : 0 < a) (hab : a ≤ b) : |
| chiStar (τ p) (X p b) (auxiliaryNoise p) - chiStar (τ p) (X p a) (auxiliaryNoise p) = |
| (((1/2 : ℝ) * ∫ u in a..b, heatFisherReal p u : ℝ) : EReal) := by |
| rw [chiStar_tail_formula p hp a ha, chiStar_tail_formula p hp b (lt_of_lt_of_le ha hab), |
| ← EReal.coe_sub] |
| congr 1 |
| have hd := intervalIntegral.integral_Ioi_sub_Ioi (fisherDeficit_tail_integrable p hp a ha) hab |
| have hg := gaussianEntropy_flow p a b ha hab |
| have hh : (∫ u in a..b, heatFisherReal p u) = |
| (∫ u in a..b, fisherDeficit p u) + (∫ u in a..b, covarianceFisher p u) := by |
| rw [← intervalIntegral.integral_add (fisherDeficit_intervalIntegrable p hp a b ha hab) |
| (covarianceFisher_intervalIntegrable p a b ha hab)] |
| apply intervalIntegral.integral_congr |
| intro u hu |
| simp only [fisherDeficit, heatFisherReal, sub_add_cancel] |
| rw [hh] |
| linarith |
| end FiniteEntropySeparation.Model |
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| end |
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| end OAI |
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